Answer:
The degree measure of ∠ACP is 112.5°
Step-by-step explanation:
The total area of the given diagram is the sum of two semicircles with arc AB and arc CB having radius R and r respectively
Where:
R = AC
r = DB
R = 2 × r
Therefore the area of the semicircles are given as follows;
For the semicircle with arc AB, we have;
Area, A₁ = π × R²/2 = π × (2×r)²/2 = 2×π×r²
For the semicircle with arc CB, we have;
Area, A₂ = π × r²/2 = 1/2×π×r²
The ratio of the two semicircles is presented in the following relation;
Therefore, the area of A₁ is four times that of A₂ or A₁ = 4 × A₂
The total area of the given diagram = A₁ + A₂ = 4 × A₂ + A₂ = 5·A₂
∴ Half of the area of the diagram, = 5·A₂/2 = 2.5·A₂ = 2.5 × 1/2×π×r² = 1.25×π×r²
The ratio of half of the diagram of the figure to the area of the semicircle with arc AB is found as follows;
/A₁ = (1.25×π×r²)/(2×π×r²) = 5/8
Therefore, the half of the diagram of the figure given by segment PAC is equivalent to 5/8 of the semicircle with arc AB
Given that the arc AB subtends an angle of 180° at the center (angle subtended by a semicircle), the arc AP will subtend 5/8×180 = 112.5°
To verify we have;
Area of a segment of a circle is presented in the following relation;
As segment PAC is 5/8 of a semicircle, it is therefore 5/(8×2) or 5/16 of the whole circle
Hence;
Therefore the degree measure of ∠ACP is 112.5°.
